We are told that DE and BC are parallel. That means triangle ADE is similar to triangle ABC — because when a line cuts two sides of a triangle and is parallel to the third side, it creates a smaller triangle that’s similar to the original one.
Similar triangles have matching angles and proportional sides. So we can set up ratios using corresponding sides.
Let’s label what we know:
In small triangle ADE:
- AD = 3
- AE = 5
In big triangle ABC:
- AB = AD + DB = 3 + x → so AB = x + 3
- AC = AE + EC = 5 + (x + 2) → so AC = x + 7
Since the triangles are similar, the ratio of corresponding sides should be equal.
So:
AD / AB = AE / AC
→ 3 / (x + 3) = 5 / (x + 7)
Wait — but in the image, they wrote:
(x + 7)/5 = (x + 3)/3
That’s actually the same thing — just flipped both sides. Let’s check:
If 3/(x+3) = 5/(x+7), then cross-multiplying gives:
3(x + 7) = 5(x + 3)
Which is exactly what was done in the image.
Now let’s solve step by step:
Step 1: Cross multiply
3(x + 7) = 5(x + 3)
Step 2: Expand both sides
Left: 3*x + 3*7 = 3x + 21
Right: 5*x + 5*3 = 5x + 15
So: 3x + 21 = 5x + 15
Step 3: Move variables to one side, constants to other
Subtract 3x from both sides:
21 = 2x + 15
Subtract 15 from both sides:
6 = 2x
Step 4: Solve for x
Divide both sides by 2:
x = 3
Let’s verify this answer makes sense.
If x = 3:
Then AB = x + 3 = 6
AC = x + 7 = 10
Small triangle: AD=3, AE=5
Big triangle: AB=6, AC=10
Ratio of sides: 3/6 = 1/2, and 5/10 = 1/2 → yes, proportional!
Also, since DE || BC, the triangles must be similar — which checks out.
Final Answer:
3
Parent Tip: Review the logic above to help your child master the concept of similar triangles.
